Max and Inflection Points of Chi-Square Distribution

Mathematical statistics uses techniques from various branches of math to prove definitively that statements regarding statistics are true. We will see how to use calculus to determine the values mentioned above of both the maximum value of the chi-square distribution, which corresponds to its mode, as well as find the inflection points of the distribution.   Before doing this, we will discuss the features of maxima and inflection points in general. We will also examine a method to calculate a maximum the inflection points. How to Calculate a Mode with Calculus For a discrete set of data, the mode is the most frequently occurring value. On a histogram of the data, this would be represented by the highest bar. Once we know the highest bar, we look at the data value that corresponds to the base for this bar. This is the mode for our data set.   The same idea is used in working with a continuous distribution. This time to find the mode, we look for the highest peak in the distribution. For a graph of this distribution, the height of the peak is a y value. This y value is called a maximum for our graph because the value is greater than any other y value. The mode is the value along the horizontal axis that corresponds to this maximum y-value.   Although we can simply look at a graph of a distribution to find the mode, there are some problems with this method. Our accuracy is only as good as our graph, and we are likely to have to estimate. Also, there may be difficulties in graphing our function. An alternate method that requires no graphing is to use calculus. The method we will use is as follows: Start with the probability density function f (x) for our distribution.  Calculate the first and second derivatives of this function: f (x) and f (x)Set this first derivative equal to zero f (x) 0.Solve for x.Plug the value(s) from the previous step into the second derivative and evaluate. If the result is negative, then we have a local maximum at the value x.Evaluate our function f (x) at all of the points x from the previous step.  Evaluate the probability density function on any endpoints of its support. So if the function has domain given by the closed interval [a,b], then evaluate the function at the endpoints a and b.The largest value in steps 6 and 7 will be the absolute maximum of the function. The x value where this maximum occurs is the mode of the distribution. Mode of the Chi-Square Distribution Now we go through the steps above to calculate the mode of the chi-square distribution with r degrees of freedom. We start with the probability density function f(x) that is displayed in the image in this article. f (x) K xr/2-1e-x/2 Here K is a constant that involves the gamma function and a power of 2. We do not need to know the specifics (however we can refer to the formula in the image for these). The first derivative of this function is given by using the product rule as well as the chain rule: f ( x ) K (r/2 - 1)xr/2-2e-x/2 - (K / 2) xr/2-1e-x/2 We set this derivative equal to zero, and factor the expression on the right-hand side: 0 K  xr/2-1e-x/2  [(r/2 - 1)x-1 - 1/2] Since the constant K, the exponential function and xr/2-1   are all nonzero, we can divide both sides of the equation by these expressions.   We then have: 0 (r/2 - 1)x-1 - 1/2 Multiply both sides of the equation by 2: 0 (r - 2)x-1 - 1 Thus 1 (r - 2)x-1 and we conclude by having x r - 2. This is the point along the horizontal axis where the mode occurs.   It indicates the x value of the peak of our chi-square distribution. How to Find an Inflection Point with Calculus Another feature of a curve deals with the way that it curves. Portions of a curve can be concave up, like an upper case U. Curves can also be concave down, and shaped like an   intersection symbol ∠©. Where the curve changes from concave down to concave up, or vice versa we have an inflection point. The second derivative of a function detects the concavity of the graph of the function. If the second derivative is positive, then the curve is concave up. If the second derivative is negative, then the curve is concave down. When the second derivative is equal to zero and the graph of the function changes concavity, we have an inflection point. In order to find the inflection points of a graph we: Calculate the second derivative of our function f (x).Set this second derivative equal to zero.Solve the equation from the previous step for x. Inflection Points for the Chi-Square Distribution Now we see how to work through the above steps for the chi-square distribution. We begin by differentiating. From the above work, we saw that the first derivative for our function is: f (x) K (r / 2 - 1) xr/2-2e-x/2 - (K / 2) xr/2-1e-x/2 We differentiate again, using the product rule twice. We have: f ( x ) K (r / 2 - 1) (r / 2 - 2)xr/2-3e-x/2 - (K / 2)(r / 2 - 1)xr/2-2e-x/2 (K / 4) xr/2-1e-x/2 - (K / 2)(r / 2 - 1) xr/2-2e-x/2 We set this equal to zero and divide both sides by Ke-x/2 0 (r/2 - 1)(r/2 - 2)xr/2-3 - (1 / 2)(r/2 - 1)xr/2-2 (1/ 4) xr/2-1 - (1/ 2)(r/2 - 1) xr/2-2 By combining like terms we have: (r/2 - 1)(r/2 - 2)xr/2-3 - (r/2 - 1)xr/2-2 (1/ 4) xr/2-1 Multiply both sides by 4x3 - r/2, this gives us: 0 (r - 2)(r - 4) - (2r - 4)x x2. The quadratic formula can now be used to solve for x. x [(2r - 4) /- [(2r - 4)2 - 4  (r - 2)(r - 4) ]1/2]/2 We expand the terms that are taken to the 1/2 power and see the following: (4r2 -16r 16) - 4 (r2 -6r 8) 8r - 16 4(2r - 4) This means that: x [(2r - 4) /- [(4(2r - 4) ]1/2]/2 (r - 2) /- [2r - 4]1/2 From this we see that there are two inflection points. Moreover, these points are symmetric about the mode of the distribution as (r - 2) is halfway between the two inflection points. Conclusion We see how both of these features are related to the number of degrees of freedom. We can use this information to help in the sketching of a chi-square distribution. We can also compare this distribution with others, such as the normal distribution. We can see that the inflection points for a chi-square distribution occur in different places than the inflection points for the normal distribution.

Human Trafficking in the US Free Essays

Sex Trafficking in the United States Imagine yourself leaving for a trip that you have been looking forward to. You get packed, board the plane, and then finally you arrive at your destination. After settling in you go explore the area. We will write a custom essay sample on Human Trafficking in the US or any similar topic only for you Order Now At this point, you are kidnapped. You cannot do anything to protect yourself or to escape. You end up being forced to strip your clothes and have your body sold. You are now an object, no longer a person. Welcome to the victims’ world of sex trafficking. Because sex trafficking is an underground business, it may be difficult to completely stop this exploitation. Knowing the signs, watching for human sex trafficking, and acting quickly can save countless lives. Sex Trafficking is any practice that involves moving people within and across local or national boundaries for the purpose of sexual exploitation (Farr 2). It is the fastest growing crime in the United States and the second largest illegal trade after the drug industry. Human Trafficking has been going on for over one hundred years. Human Sex Trafficking happens all over the world, including the United States. There are approximately twenty-seven million victims of sex trafficking worldwide; three undred thousand of them are women and children bought and sold yearly here in the U. S. Of the three hundred thousand sex slaves sold in the US, 25% are forced into the business by parents or family members and 75% are former runaways. Each year, 1. 7 million children run away from home. From that number, 90% will be approached by traffickers within 48 hours. Many of which do not make it out the industry alive. In the state of Texas alone, there are only 99 known survivors in the last 20 years whoVe managed to escape sexual slavery (Preda. org). Victims of sexual slavery are not all kidnapped, majority are tricked into the business. Most of the women trafficked into brothels posing as massage businesses are not U. S. citizens, and many do not have documented status. Therefore, traffickers use the threat of deportation to maintain control of immigrant women. Without legal status, the women frequently fear and distrust police or government authorities. Immigrant women are vulnerable due to language barriers, unfamiliarity with their legal rights in the US, and/or the lack of a local support network. Others can be women with financial need or in debt leave the women vulnerable to recruiters, who appear to be offering legal Jobs. Sex trafficking thrives because it is low in risk and high in pay off. The United States FBI estimates about three thousand Russian mobsters control gangs in American cities that involve forced prostitution (Stoecker, Shelly 14). These groups generate 7 billion dollars yearly in the United States. Since there are currently no consistent or accurate ways of tracking these crimes, statistics may vary. Because this is a large and very detailed business, many people are involved throughout the process. Everyone involved has a role to play to ensure the process is done quickly and smoothly. The recruiter finds and brings the victims into the industry usually by force or deception. The recruiter then sells the women to brokers or directly to employers. Brokers are the â€Å"middleman†; they buy the women from recruiters and sell them to employers, those who own brothels or bars. If the women are being transported from overseas, a contractor organizes the transaction. Next an Employment or Travel agent is needed. Their main purpose is to arrange a â€Å"legitimate† Job and Job description or a† legitimate† trip. A document theft or forger obtains all the legal documentation needed to travel from country to country. The transporter travels with the women to each destination and delivers them to the recruiter. Who then sells them to the employers. The employers provide the women with a place to live and work; telling them of the working conditions, living arrangements, and lifestyle. A large number of employers are bar or club owners, while a very small percentage of them are street pimps. For large establishments an enforcer serves as security for the place of business (Farr 63). The life of a sex slave can be compared to an animal caged in a zoo. You are brought out only to do your trick, and then locked in a cage again. Victims, usually ages 11-17, are chained to a bed or confined in small living quarter’s majority of the day until they are fed, usually something light to keep their weight down. Then they are given ice-cold showers to reduce the swelling on their bodies. Attendants then cover-up their bruises from the night before, put make-up on the girls, and then present them to another group of men for more abuse and profit. The women usually serve 5 to 30 men a night. Many of these victims turn to drugs as a solace for the life they have been brought into. Drugs and beatings numb their capacity for thoughts of escape and further iscourage the energy or alertness required to act on their desire for freedom. Many never make it out of the industry alive (preda. org). Stopping Sex Trafficking is harder than it may seem. There are signs of human sex trafficking that everyone should be aware of. Visible indicators may include: Heavvy security at the commercial establishment including barred windows, locked doors, isolated location, and electronic surveillance. Women are never seen leaving the premises unless escorted. Victims live at the same premises as ‘the brothel or work site, or are driven between uarters and â€Å"work† by a guard. Victims are kept under surveillance when taken to a doctor, hospital or clinic for treatment; trafficker may act as a translator. High foot traffic especially for brothels where there may be trafficked women indicated often by a stream of men arriving and leaving the premises. Physical signs of a person being trafficked include: malnutrition, dehydration or poor personal hygiene; sexually transmitted diseases; signs of rape or sexual abuse; bruising, broken bones, or other signs of untreated medical problems; critical illnesses including diabetes, cancer or eart disease; post-traumatic stress or psychological disorders (humantraffcking. org). How to cite Human Trafficking in the US, Papers

Criminal Procedure for Fiduciary or Confidential Relationship

Question: Discuss about theCriminal Procedure for Fiduciary or Confidential Relationship. Answer: Generally, a lawyer and client maintain a fiduciary or confidential relationship between each other as stated in decision of Descteaux v. Mierzwinski [1982] 1 SCR 860 where the Court recognise the communication of client and solicitor as confidential and fundamental right. However the response to the problem deals with an exception of the privilege of solicitor client that is (1) Where nature of lawful guidance is not pursued; (2) Where no intentions to make any confidential communication; (3) Communication is in nature of fostering any illegitimate conduct of client (Lefstein, 2007). The primary object of the privilege communication is to build a privacy area in order to preserve confidentiality without accusatorial meddling in litigation and without distress of early discovery of privileged communication. In support of response to the problem, the court is not bound to maintain the privileged communication in any circumstances (Griffiths, 2014). The general rule of section 4(3) of the Canada Evidence Act state that that no spouse communication can be obliged to testify in any criminal matter as ruled under R. v. Schell (2004), 188 C.C.C. (3d) 254, 20 C.R. (6th) 1 (Alta. C.A.)., where the Court held that spousal privileged communication deals with a vital principle of the protection of marital harmony. It is a fundamental principle that no spousal privileged communication is protected by the court unless it is declared or asserted. Such spousal privileged communication completely depends on the assertion of the spouse. In order to establish the principle of the spousal privileged, the counselor of spouse has to declare a right of spousal immunity and must distinguish between the scope of spousal testimonial privileged and spousal privilege. (Posner, 2008) The following conversation are protected in spousal privileged are The communication either in the form of oralorwritten. Thecommunicationmustbein nature ofaconfidentialcharacter. Any communication discloses any criminal liability over a spouse. Thecommunicationmustcreateina permanentassurancethatitneverbedisclosed to any other (Martin, 2001). Informer privileged has been originated by the leading case of 19th century Marks v Beyfus (1890), 25 QBD 494. The case deals with a significant factor to protect the informer because informer plays a vital role in the protection of material evidences during investigation. The verdicts were that informer is classified under special criteria of class privilege where absolute protection is provided to the informer. Even the court cannot exercise its discretionary power to waive the protection to informer privilege (Haig, Raikes MacMillan, 2014). The specific category of privilege is provided to police informer. The police informer plays a vital role in concealing the important sources or information or networks of the police and also provides aid to police in any investigation process of the case. The court examines the legal application in the case of R. v. Named Person B, 2013 SCC 9 to protect the police informer in order to assure the confidentiality in any official matters. The court considers the view that police informer deemed to be a hidden agent of the police who ultimately work in interest of public. The two ways by which Crown Prosecutor can invoke informer privilege are The crown prosecutor can invoke the application of section 37 of theCanada Evidence Act. The case R v Meuckon (1990), 57 CCC (3d) 193 (BCCA) states that assertion can be personally claimed by the prosecutor. The appeal of acquittal can be requested by the crown prosecutor by asserting the court that opposite counsel does not support the verdicts and it does not provide any material evidences to the case. If the disclosure is not feasible then in such condition the crown is available with an option to stay the court proceedings under section579of theCriminal Codeon a ground of an unexceptional circumstance made out in a case where the justice can be preserved and maintained in favour of informer privileges. Such option availed the opportunity to re commence the litigation in future. The only exception to informer privilege is that Innocence at Stake. The exception permits the breach of privileged on a factor raise on reasonable doubt. Under this exception, the informer privilege and solicitor and client privileged are not offended by any breach in a privileged communication. The exception to innocence at stake specified in case of R v. Leipert (Gillers, 2014). References Gillers, S. (2014).Regulation of Lawyers: Problems of Law and Ethics. Wolters Kluwer Law Business. Lefstein, N. (2007). The Criminal Defendant Who Proposes Perjury: Rethinking the Defense Lawyer's Dilemma.Hofstra L. Rev.,6, 665. Griffiths, C. (2014).Canadian Criminal Justice: A Primer, 5e(Vol. 5). Nelson Education. Posner, R. A. (2008). Privacy, surveillance, and law.The University of Chicago Law Review,75(1), 245-260. Haig, J., Raikes, G., MacMillan, V. (2014).Cites sources: An APA documentation guide. Nelson Education. Martin, D. L. (2001). Lessons about justice from the laboratory of wrongful convictions: Tunnel vision, the construction of guilt and informer evidence.UMKC L. Rev.,70, 847.

10 Argon Facts - Ar or Atomic Number 18

10 Argon Facts - Ar or Atomic Number 18 Argon is atomic number 18 on the periodic table, with the element symbol Ar. Here is a collection of useful and interesting argon element facts. 10 Argon Facts Argon is a colorless, flavorless, odorless noble gas. Unlike some other gases, it remains colorless even in liquid and solid form. It is nonflammable and nontoxic. However, since argon is 38% more dense than air, it present an asphyxiation risk because it can displace oxygenated air in enclosed spaces.The element symbol for argon used to be A. In 1957, the International Union of Pure and Applied Chemistry (IUPAC) changed argons symbol to Ar and mendeleviums symbol from Mv to Md.Argon was the first discovered noble gas. Henry Cavendish had suspected the elements existence in 1785 from his examination of samples of air.  Independent research by H.F. Newall and W.N. Hartley in 1882 revealed a spectral line that could not be assigned to any known element.  The element was isolated and officially discovered in air by Lord Rayleigh and William Ramsay in 1894. Rayleigh and Ramsay removed the nitrogen, oxygen, water, and carbon dioxide and examined the remaining gas. Although other eleme nts were present in the residue of air, they accounted for very little of the total mass of the sample. The element name argon comes from the Greek word argos, which means inactive. This refers to the elements resistance to forming chemical bonds.Argon is considered to be chemically inert at room temperature and pressure.Most of the argon on Earth comes from the radioactive decay of potassium-40 into argon-40. Over 99% of the argon on earth consists of the isotope Ar-40.The most abundant isotope of argon in the universe is argon-36, which is made when stars with a mass about 11 times greater than the Sun are in their silicon-burning phase. In this phase, an alpha particle (helium nucleus) is added to a silicon-32 nucleus to make sulfur-34, which adds an alpha particle to become argon-36. Some of the argon-36 adds an alpha particle to become calcium-40. In the universe, argon is quite rare.Argon is the most abundant noble gas. It accounts for about 0.94% of the Earths atmosphere and about 1.6% of the Martian atmosphere. The thin atmosphere of the planet Mercury is about 70% argon.  No t counting water vapor, argon is the third most abundant gas in the Earths atmosphere, after nitrogen and oxygen. It is produced from fractional distillation of liquid air. In all cases, the most abundant isotope of argon on the planets is Ar-40. Argon has many uses. Its found in laser, plasma balls, light bulbs, rocket propellant, and glow tubes. Its used as a protective gas for welding, storing sensitive chemicals, and protecting materials. Sometimes pressurized argon is used as a propellant in aerosol cans. Argon-39 radioisotope dating is used to date the age of ground water and ice core samples. Liquid argon is used in cryosurgery, to destroy cancerous tissue. Argon plasma beams and laser beams are also used in medicine. Argon may be used to make a breathing mix called Argox to help remove dissolved nitrogen from the blood during decompression, as from deep-sea diving. Liquid argon is used in scientific experiments, including neutrino experiments and dark matter searches. Although argon is an abundant element, it has no known biological functions.Argon emits a blue-violet glow when it is excited. Argon lasers exhibit a characteristic blue-green glow.Because noble gas atoms have a complete valence electron shell, they are not very reactive. Argon does not readily form compounds. No stable compounds are known at room temperature and pressure, although argon fluorohydride (HArF) has been observed at temperatures below 17K. Argon forms clathrates with water. Ions, such as ArH, and complexes in the excited state, such as ArF, have been seen. Scientists predict stable argon compounds should exist, although they have not yet been synthesized. Argon Atomic Data Name Argon Symbol Ar Atomic Number 18 Atomic Mass 39.948 Melting Point 83.81 K (189.34 C, 308.81 F) Boiling Point 87.302 K (185.848 C, 302.526 F) Density 1.784 grams per cubic centimeter Phase gas Element Group noble gas, group 18 Element Period 3 Oxidation Number 0 Approximate Cost 50 cents for 100 grams Electron Configuration 1s22s22p63s23p6 Crystal Structure face-entered cubic (fcc) Phase at STP gas Oxidation State 0 Electronegativity no value on the Pauling scale Bonus Argon Joke Why dont I tell chemistry jokes? All the good ones argon!

young offenders essays

young offenders essays In recent years, in Canada, we have seen a gradual reduction in the crime-rate. However, every day on the radio and on TV, we see and hear of another armed robbery, another sexual assault, another drug bust, or another brutal murder. This is scary because it affects us all. We are all potential victims; we are all susceptible to these horrible crimes. Even scarier is that more and more of these crimes, the violent ones, involve young offenders. We are hearing more and more about young offenders, youth between the ages of twelve and eighteen, who are stabbing people on school grounds, sexually assaulting others, and murdering their peers. Recently, we heard the story of a young boy, only six years old, who murdered a fellow classmate in a Michigan elementary school. A few years ago, we heard of the two young boys in England who murdered a boy who was under the age of three years. The dealing of narcotics on the school ground, be it elementary or high school, is increasing. Gradually, it seems that people from every age group are becoming victims, and people from every age group are becoming offenders. Because of space limitations, this paper will deal only with a few young offenders issues. In addition, only a few aspects of the Young Offenders Act will be dealt with. The Young Offenders Act is the federal law for young people charged with crimes. Prior to April 2, 1984, the Young Offenders Act did not exist in Canada. Instead, the Juvenile Delinquents Act (which had been used since 1908) was used to deal with young offenders. The main philosophy of this latter Act was to deal with the welfare of the child. In April 1984, the Young Offenders Act was enacted and one of the major changes that took place was in the philosophy used to deal with young offenders. The Young Offenders Act recognizes that in some situations, a young offender may be deemed (by the trial judge) to be beyond rehabilitation, and a le...

Fundamentals of Financial Management Speech or Presentation

Fundamentals of Financial Management - Speech or Presentation Example tio has decreased in 2010 standing at 1.9:1 and as compared to the last year which was 2.5:1 and the quick ratio which is 0.69:1 in 2010 as compared to 1.09:1 in 2009, but the other ratios shows the company has improved a lot in managing its financial position as well as performance from that of last year. Current Ratio – A current ratio measures the relationship of current assets with current liabilities. This is used to analyze the solvency of a firm. Higher current ratio measures the firm’s higher rate of solvency. Acid-Test Ratio (Quick Ratio) - This ratio is used to analyze the liquidity position of a company. It helps an organization to meet the short term obligation with the liquid assets it has. A firm is evaluated to be in better condition when the quick ratio is higher. Average Age of Receivables – While reviewing a firm’s financial health this is important because it is compared to the credit and collection policy and the industry in which the firm operates. It is countable in terms of days. The analysis shows better result if number of days is higher. Inventory Turnover Ratio – This ratio shows number of times the inventory of a company is being sold and replaced over a certain period of time. Generally a low turnover indicates bad sign for a company. Times Interest Earned – This is a tool that is used to measure an ability of a company to meet its obligations that is related to debt. If any company fails to meet its obligations then it will be in a position of bankruptcy (Ramachandran & Kakani, 2007). Changes can occur in both the ways, positive or negative. The reason behind the lower current ratio and acid test ratio can be similar and that is cash shortage. Basically the poor credit policy and the collection process of the company can be blamed for higher average age of receivables. Again an inventory turnover ratio implies that the company must have a tight control over the purchasing and simultaneously it also refers that due to

Seigel v. Merrill Lynch Case Analysis Study Example | Topics and Well Written Essays - 500 words

Seigel v. Merrill Lynch Analysis - Case Study Example As a matter of fact, Seigel was obliged to make these payments anyway. There was no reason for him to stop making these payments. It is also a fact that he had given instructions to Merrill Lynch to stop payment against the checks issued by him to the casinos against gambling chips. However, the court ruled that since Seigel had not suffered any actual loss and the checks were payments rightfully made in view of his gambling dues, Merrill Lynch could not be convicted for any wrong doing. Prima facie, Merrill Lynch erred in not adhering to the instructions to stop payments against the checks. It was a mistake on the part of Merrill Lynch to pass the checks after having received instructions from their client, Seigel, to stop making payments against these checks. In fact, Merrill Lynch did stop payments against many checks issued to the casinos by Seigel. However, they did not fully adhere to instructions and cleared certain checks amounting to $143,000 issued by Seigel to these casinos. The court's view that since Seigel abrogated his rights to these monies after losing them to the casinos, he could not seek legal reprieve even if Merrill Lynch had overlooked his instructions to stop payment against these checks appear proper and fair.